SECTION 17.2 Summary

• Software fits the data to the model, producing estimates $b_0$ and $b_1$ of the parameters ${\beta }_0$ and ${\beta }_1$. Software also produces standard errors for these estimates.

• A level $C$ confidence interval for the slope ${\beta }_1$ is

  $b_1\pm z^{*}{\text{SE}}_{b_1}$

  A level $C$ confidence interval for the odds ratio $e^{{\beta }_1}$ is obtained by transforming the confidence interval for the slope,

  $\left(e^{b_1-z^{*}{\text{SE}}_{b_1}},e^{b_1+z^{*}{\text{SE}}_{b_1}}\right)$

  In these expressions, $z*$ is the standard Normal critical value with area $C$ between $-z*$ and $z*$.

• The null hypothesis that $x$ does not help predict $p$ in the logistic regression model is $H_0:{\beta }_1=0$ or $H_0:e^{{\beta }_1}=1$ in terms of the odds ratio. To test this hypothesis, compute the test statistic

  $X^2={\left({\displaystyle \frac{b_1}{{\text{SE}}_{b_1}}}\right)}^2$

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  In terms of a random variable ${\chi }^2$ having a ${\chi }^2$ distribution with 1 degree of freedom, the $P$-value for a test of $H_0$ against $H_a:{\beta }_1\ne 0$ is approximately $P\left({\chi }^2\ge X^2\right)$.